CC BY
2Luigi De Cesare and Andrea Di Liddo
Faculty of Economics,
University of Foggia,
Largo Paga Giovanni Paolo II, 1, 71100 Foggia, Italy E-mail: l.decesare@unifg.it E-mail: a.diliddo@unifg.it
Abstract We consider a number of firms who sell a non differentiated product through periodic subscriptions. Firms select price and advertising strategies maximizing their profits and buyers behave differently according to their status: new buyer or renewer. A discrete dynamic game is proposed together with a numerical alghorithm based on the simulated annealing technique.
Keywords: Price and advertising models, Marketing models, Dynamic games, Simulated annealing
1. Introduction
In this paper we consider a number I of firms (e.g. publishers) who sell a non differentiated product (e.g. a journal) through periodic subscriptions.
The sales process is considered discrete in time. A customer pays at time t the price Pi,t-1 to one of the producers i and receives 0 items at times t,t +1,t — 1 + 0. It makes sense that everyone subscribes at most one product.The potential customers are convinced to subscribe through advertising performed by firms.
Moreover previous subscribers influence potential adopters transmitting their experience (interpersonal contacts) and buyers react also to the prices. Firms select price and advertising strategies maximizing the present values of their profits and buyers behave differently according to their status: new buyer or renewer. The adopters dynamics is described by a system of difference equations of order 0.
As usual we consider quadratic advertising costs to model decreasing returns.
In Section 3 a two stages game is considered: two symmetric firms compete and the game is explicitly solved by backward induction. In this section the advertising performed by one firm increases also the sales of the other one.
In Section 3 we discuss a simple example in which the advertising performed by one firm increases the sales of itself but decreases the sales of the competitor. The one stage case is solved explicitly while the two stage game is studied through a numerical example.
Section 4 contains the body of the paper: the price-advertising multistage, multiplayers game is proposed and feedback equilibria are characterized by means of the Bellman equations. Finally a numerical alghorithm is proposed based on the simulated annealing technique and a numerical experiment is showed.
2. A two stages symmetric duopoly
In this section we consider two firms who sell a non differentiated product (e.g. a journal) through periodic subscriptions to a population of N potential subscribers. At time t = 0, the firm i, i = 1, 2 launches an advertising campaign whose cost is
OaYio. As a consequence yi1 individuals pay the subscription fee at the time t = 1 and at the same time they receive one item of the product i.
Therefore at the time t = 1 the population is divided into two classes: subscribers *1 = yii + y21 and potential adopters N — x1.
Moreover at the time t = 1 the two firms spend an amount caY21 to advertise their product further.
At the time t = 2 a fraction yi2 of potential adopters N — x1 and a fraction zi2 of previous subscribers buy the product i.
Each firm chooses the advertising expenditure to maximize the net present value of its profits.
A dynamic discrete game arises and we compute the Nash equilibria by backward induction.
The subscribers dynamics is given by
’ Vn = ^(/?7io + (1 - P)l2o)N V21 = I((l - /?)710 + /3l20)N
and
X1 = y11 + y21 ’ V12 = 711 + (1 - P)l2l){N - Xl)
V22 = ^((1 -/3)711 + Pl2l){N — Xi)
Z12 = ff(Ayn + (1 — A)y21)
^ Z22 = ^((1 — A)y21 + Ayu)
We assume that 0 < 7^ < 1, i = 1,2, j = 0,1. (3 g]|, 1[ is the advertising
efficiency; aA is the renewal rate while <r(1 — A) is the cross renewal rate, where
0 < A < 1, 0 < a < 1.
At time t = 1 the two firms choose advertising to maximize their last stage payoff.
max Ln max L21
0<7n<1 0<72i<1
where
L11 = (1 — cp)(y12 + z12) — caY21
L21 = (1 — cp)(V22 + z22) — caY21
Here cp is the production cost of a unit of the item (the same for both firms) and we assume that customers pay 1 euro for one subscription.
This last stage game has a unique Nash equilibrium
* * . (1 a/3(N-x 1)
7n = 721 = mm 1,--------------------
where
1 - cp
a =------------
2ca
Note that 7*1 = 7I1 is increasing with respect to the efficiency parameter 3 and with respect to the total number of potential adopters N - x1 ; it is decreasing with respect to the unit advertising cost ca and the unit production cost cp.
At time t = 0 the two firms foresee the last stage equilibrium and choose advertising to maximize their two stage payoff. The game has the unique Nash equilibrium
* * (n . (, a/3N (a(3N((3 — 2) + 2 (a + 1))
7io = 720 = max 0, mm 1, -
N 2a232(3 — 2) +4 Let us illustrate this result through the following numerical example.
Assume that N = 1000, 3 = 0.6, cp = 0, ca = 500, A = 0.8.
Then the last stage of the game has the unique Nash equilibrium
* _ * _ 3000 - XI
7n _ 721 _ 5000
At time t = 0 the game has the unique Nash equilibrium
3(25a +13) 14
if a < —
710 = 720
53 75
if a > — “ 75
Note that the optimal advertising effort at time t = 0 is increasing with respect to the rate of renewals a.
3. A symmetric one stage non linear game
In the previous section we assumed that the advertising performed by the firm i increases the sales of both firms i and j, although with a different intensity.
In this section we discuss a simple example in which the advertising performed by the firm i increases the sales of itself but decreases the sales of the competitor j. We solve explicitly the one stage case and discuss the two stage game through a numerical example.
3.1. The one stage game
At time t = 0, the firm i, i = 1, 2 launches an advertising campaign whose cost is caY2. As a consequence yi individuals pay the subscription fee at the time t = 1 and at the same time they receive one item of the product i.
Therefore at the time t = 1 the population is divided into two classes: subscribers x1 = y1 + y2 and potential adopters N - x1 .
Each firm chooses the advertising expenditure to maximize the net present value of its profits.
The subscribers dynamics is given by
yi = itopl (/?7l + (1 _ 3yn)N
y2 = (I±3pl((1 - 3)lx + {3l2)N
At time t = 0 the two firms choose advertising to maximize their payoff.
max L1 max L2
0<7i<1 0<72<1
where
L1 = (1 — cp)y1 — caY2 L2 = (1 — cp)y22 — caYl
L1 and L2 are second degree polynomial with respect to Y1 and Y2 so that we can explicitly solve the game by standard although cumbersome calculations.
Let
1 — cp Na/3 a.N(f3—l)
a ~ 2ca ° ~ 2 - aJV ^ ~ 2{a(3N - 1)
2
a if 0 < Na <
3+1
Y1 = Y2 =
1 if < Na
/3+1 “
2 2 2
Moreover if--------- < Na < -------- and /3 > - then the game has other two further
4/3-1 /3 + 1 ^ 3 S
Nash equilibria,
Y1 = v y2 = 1 Y1 = 1 Y2 = v
3.2. The two stages game
In this section we use the same notations of Section 2 to discuss the two stages game assuming that the advertising performed by the firm i increases the sales of itself but decreases the sales of the competitor j.
As a consequence the dynamics of subscriptions is now given by
yn = (1+7ir7a°)(/?7io + (1 - /3)720)^
y21 = (1+72r7lo)((l - /3)7io + /3720)AT
x1 = y11 + y21
y12 = (1+7V72i)(/37h + (1 - /3)72i)(W - xi)
V22 = (1+^pll}((l - /3)711 + /3721)(W - xx)
Z12 = a(Ayu + (1 — A)yn)
Z22 = a((1 — A)yu + Ayn)
As in the Section 2 the two firms choose advertising expenditure to maximize their profits.
We cannot obtain an explicit solution by we illustrate a numerical example solved by backward induction.
Assume that N = 1000, 3 = 0.6, cp = 0, ca = 500, A = 0.8.
Then the last stage of the game has the unique Nash equilibrium
* _ * _ 3(1000 -xi)
7n - 721 - 5(1000 + xi)
At time t = 0 the two firms foresee the last stage equilibrium and choose advertising
to maximize their two stage payoff.
Since y*1 = Y21 are rational functions of x1, it is not possible to compute explicitly the equilibria at the initial stage. For the above setted values of the parameters we prove that
the solution of first order conditions if a < —
29
if a > —
“ 29
4. The multistage price and advertising oligopoly model
From experience gained in the previous sections we can more effectively introduce a more general model in which the price of the products are no longer given exogenously.
4.1. The subscriptions dynamics
Let us consider a number of firms I (e.g. a publishers) who sell a non differentiated product (e.g. a journal) through periodic subscriptions. The sales process is considered discrete in time and we assume that decisions are taken at time t e {0,1,..., T — 1}, where T e IN. Let 0 be the length of a subscription. A customer pays at time t the price pi,t-1 to one of the producers i and receives 0 items at times t,t +1, ...,t — 1 + 0. It makes sense that everyone subscribes at most one product. The potential customers are convinced to subscribe through advertising performed by firms. Moreover previous subscribers influence potential adopters transmitting their experience (interpersonal contacts) and buyers react also to the prices. In this framework firms compete selecting price and advertising strategies maximizing the present values of their net earnings and buyers behave differently according to their status: new buyer or renewer. We assume that the total population of potential subscribers, denoted by N, is constant in time. We define the following classes:
— yiit number of new subscribers of the i-product at time t,
— zi,t number of subscription renewals of the i-product at time t,
— xi,t total number of subscribers of the i-product at time t,
— wi,t total number of subscriptions of the i-product at time interval [t,t + 1],
— xt = El=1 xi, t total number of subscribers at time t.
Let us denote by Yi,t, the quantities (normalized to one) of the advertising performed at time t by firm i. The effectiveness of the advertising is measured through a function gi which we assume is increasing and concave in its arguments to incorporate
decreasing advertising returns. Furthermore people are convinced to buy the product through interpersonal contacts with previous adopters (word-of-mouth). This effect is modelled by parameter ki,j which represents the effect of word-of-mouth to convince a non adopter to buy the product i due to the interpersonal contacts between adopters of the product j with the non adopters, i,j = 1,...,I. We assume that reaction to prices is modelled by a price-response function qi.(p1,t,...,pi,t) which is increasing with respect to pj,t, j = i.
Following the Bass model (Bass, 1969), we assume that new subscribers dynamics is given by
yi,t+1 = qi(p1,t,...,pi,t) ki,jxj,t + gi(n,t,...,n,t)^ (N — xt)
We assume that a part Ai < 1 of the expired subscriptions of the product i are renewed. Moreover the coefficient Ai depends on the current price pi,t and the advertising level Yi,t. The total number of subscription renewals of the i-product at time t is:
zi,t+1 = Ai(pi,t , 7i,t)wi,t+1-0 .
Hence the total number of subscriptions at time interval [t, t +1] of i-product is
Wi,t+1 = yi,t+1 + zi,t+1. while the total number of subscribers at time t is:
xi,t+1 = xi,t + Wi,t+1 — Wi,t+1-9.
Finally the adopters dynamics is described by the following system of difference equations
x1,t+1 = x1,t + q1 (p1,t,...,pi,t) k1,j xj,t + g1(Y1,t,...,Yi,t)j (N — xt)
— (1 — A1)w1,t+1-e
W1,t+1 = q1(p1,t,... ,pi,t) k1,j xj,t + g1(71,t,...,7i,t)^ (N — xt)
+A1W1,t+1-e
. (1)
xi,t+1 = xi,t + qi(p1,t,...,pi,t) ki,jxj,t + gi(71,t,...,7i,t)^ (N — xt)
— (1 — Ai )wi,t+1-e
wi,t+1 = qi(p1,t,. ..,pi,t) ki,jxj,t + gi(71,t,...,7i,t)^ (N — xt)
+Ai wi,t+1-e
with the initial conditions
Xi,o = 0, i = 1,...,I
wi,-(e-i) = Wi,-(e-2) = .... = Wi,o =0, i = 1,...,I.
Furthermore, we suppose that the per unit advertising cost, the unit production cost, the discount factor remain constant during all the time horizon and we denote them by ci,a, cip and r, respectively. As usual we consider quadratic advertising costs to model decreasing returns.
We assume that buyers pay the subscription in advance, thus the profit of the
i-th firm is given by
T-1
Li e rt(e r (Pi,t wi,t+1 — ci,pxi,t+1) — ci,al1}t) — &i(T)
t=0
where
{T-2+e
ci,p e-r(t+1)xi,t+i if 9 > 2
t=T
0 if 9 = 1.
The term ^i(T) is due to the production costs for the remaining subscribers at the end of the planning period.
4.2. The multistage game
We are interested in finding an I -tuple of strategies {(pit.,%,.) & [0, +^[T x[0,1]T ,i = 1,...,I} that constitute a feedback Nash equilibrium of an I -person T -stage discretetime infinite dynamic game.
The adopter dynamics is given by a system of difference equations with a (9 — 1)-delay in the state variable w. This system can be reduced to a first order difference equation system adding 9 — 1 dummy variables and 9 — 1 equations for every player. We define
e-1
pi,t = wi,t-(e-i)
P1,t = wi,t-i Hence we have the following equations
Pe-1 = pe-2 pi,t+1 ~ pi,t
P1,t+1 = wi,t
Let Yi,t = (xijt,wi,t,p11t,...,pe-x) the vector of state variables at time t for the i-firm and ^ , t = (pi , t, y,t) the strategies of the i-firm at time t. We rewrite the state equations as
Yi,t+1 = fi^ C1, t,... &,t)
where fi can be easily obtained substituting dummy variables pi in (1) and Yt = (Y1t,..., Yr,t). In this form the adopter dynamics is a first order difference system
with (9 + 1)I state variables. For the sake of simplicity, we put the discount factor r = 0. We observe that the i-firm profit at stage t < T is
gi,t (Yt, &1,t, . . . ,&i-1,t,pi,t, Yi,U &i+1,t, . .., &I,t) • = pi,t wi,t+1 — ci,pxi,t+1 — ci,aYi,t.
We set
&i (Yt, &1,t, . . . , &i-1,t ,p, Y, &i+1,t, . . . , &I,t) •
qi(Pl,t,... ,pi-l,t,p,pi+l,t,.. .pi,t)-
^2 ki,j xj,t + 9i(Yl,t,..., Yi-1,u Y, Yi+l,u... Yi,t) I (N — xt)
Vj=1 )
then from (1) we have
9i,t(Yt, &l,t, . .., &i-l,t,p, Y,&i+l,t, .. ., &I,t)
= p(&i + Ai(Pi- 1) — ci,p(xi,t + &i — (1 — \)Pe- 1) — ci,a Y2
Applying standard results (see Basar and Oldser, 1999) for feedback Nash equilibria, we have that a set of strategies (£* t,...,&I t),t = 0,...,T — 1 is a feedback Nash equilibrium if and only if there exist functions
Vi , t • IR(e+1)I i—— HR, i — 1, ...,I
such that the following recursive relations are satisfied:
Vi , t (Y ) = —$i(T) i = 1,...,I
Vi , t(Y )= 9i, t.(Y,Cl, t ,...,&i--l, t ,p,Y,&*i+l, t,...,&I , t) +
(p,Y)£[0, + ro[x[0,1] ’ ’ ^ ’
Vi^Ul^^l^... ^i-lt^r/^i+lt. ..,&I,t),...
..., fI(^ Cl,t,..., &i-l,t,P, Y, €i+l,u ..., &I,t)) =
gi,t(Y, &,t,..., &,t) + Vi,t+l(fl(Y, Cl,t,..., &,t),..., fI (Y, Cl,t,..., tI,t))
4.3. Numerical solutions
The value functions also provide the Nash equilibrium profit for all firms.
Due to the non linearity of state equations and a general form of profit functions, it is very difficult to derive analytical information from previous relations. Therefore a numerical solution of Nash equilibrium can be obtained implementing an iterative algorithm. For simplicity we consider the case of two players. First of all we build a mesh of the state space which is a subset of IR2(e+1).
For each stage, going downward from T-1 to 0, and at every mesh-node, we perform the following steps:
1. Choose an initial value for control parameters of firm i =1.
2. Execute the following steps ...
2.1 Compute the best reaction, +1, of firm i = 2, if the firm 1 play &k.
2.2 Compute the best reaction, &k+1, of firm i = 1, if the firm 2 play &!^+1.
... repeat step 2.1 and 2.2 until suitable tolerance is reached.
For 9 = 2, the best reactions are computed solving the following optimization problems.
For firm 1:
max p (&l(Y,p,Y,S2,)+\lPl) — cl,p(xl + &i(Y,p,y,£2)
(p,7)e[0, + ^[x[0,1j
— (1 — Al)pl) — cl,aY2 + Vl,n+l(fl(Y,p, Y, e2), f2(Y,p, Y, $))
For firm 2:
max p (&2(Y,p,Y,SX) + A2P2) — Q,p(x2 + &2(Y,p,Y,£!i)
(p,7)^[0,+^[x[0,1j
— (1 — A2)p2) — ciaY2 + V-,n+l(fl(Y,(* ,p,Y),f2(Y,ef ),p,y)
To compute the global maximum at every stage, the simulated annealing algorithm can be used (see Kirkpatrick et al., 1983). This is an heuristic method for global optimization which requires no particular assumptions on objective function (i.e. convexity).
It is known that the existence of a Nash equilibrium is still an open problem. If multiple equilibria exist then the numerical method finds only one of them. Also the convergence of previous iterative method remains an open problem.
We assume that the price-response function q1(p1,p2) for the first product is increasing with respect to p2 and that it is decreasing with respect to p1. Furthermore 0 < q1 (p1,p2) < 1. The analogous properties hold for the price-response function q2 of the second product. We choose price-response functions as follows
[ ql(pl,p2) = exp(—«ipi)^i(p2 — pi)
Iq2(pl,p2) = exp( —a2p2)^2(pl — p2)
where ai are positive constants and i are increasing functions. So, potential consumers react to the price of the product they are going to buy but also they react to the difference between the prices of the two products. In the simulation performed, we set
{fi(PuP2) = ^ (arctan(p2 -Pl) +
f2(pi,p2) = ^ (arctan(pi -p2) +
Moreover we choose linear advertising effectiveness functions gi
( gl(Yl,Y2) = al,lYl + al,2Y2
I g2(Yl,Y2) = a2,lYl + a2,2Y2
where ai,j are positive constants. We assume that coefficients Ai are constant.
The following figures illustrate a numerical experiment obtained by the techniques described above and using the values of the parameters given in Table 1.
Price (Firm 1)
Table1. Simulation n. 1
Q-i Xi ci,p Ci,a ki,i ki,2 ai,i ai,2
firm 1 0.5 0.9 1.0 0.1 0.050 0.002 0.30 0.05
firm 2 0.5 0.8 0.9 0.1 0.001 0.040 0.05 0.30
Figure1. Simulation n.1: Advertising-price strategies
References
Bass, F. M. (1969). A new product growth model for consumer durables. Management Science, 15, 215-227.
Basar, T. and G. J. Oldser (1999). Dynamic Noncooperative Game Theory. Siam. Philadelphia.
S. Kirkpatrick, S., C. D. Gelatt and M. P. Vecchi (1983). Optimization by Simulated Annealing. Science, Vol. 220, No. 4598, 671-680.
